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Let G be a semisimple complex algebraic group with a simple Lie algebra , and let ℳ G 0 denote the moduli stack of topologically trivial stable G -bundles on a smooth projective curve C. Fix a theta characteristic κ on C which is even in case dim is odd. We show that there is a nonempty Zariski open substack κ of ℳ G 0 such that H i (C , ad (E G) ⊗ κ) = 0 , i = 1 , 2 , for all E G ∈ κ . It is shown that any such E G has a canonical connection. It is also shown that the tangent bundle T U κ has a natural splitting, where U κ is the restriction of κ to the semi-stable locus. We also produce an isomorphism between two naturally occurring Ω M G r s 1 -torsors on the moduli space of regularly stable M G r s . [ABSTRACT FROM AUTHOR]

We consider for structure groups SU (n) ⊂ SL (n , ℂ) a densely defined toric structure on the moduli space of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces. [ABSTRACT FROM AUTHOR]

First we survey and explain the strategy of some recent results that construct holomorphic sl (2 , ℂ) -differential systems over some Riemann surfaces Σ g of genus g ≥ 2 , satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl (2 , ℂ) -systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup Γ ⊂ SL (2 , ℂ) as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL (2 , ℂ) / Γ , where Γ ⊂ SL (2 , ℂ) is a cocompact lattice, that do not factor through any elliptic curve [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL (2 , ℂ) , J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. Then we prove that when M is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over ℂ obtained from the category of holomorphic connections on étale trivial holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If ϖ (respectively, Θ) stands for the affine group scheme over ℂ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : (Θ) → (ϖ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals. [ABSTRACT FROM AUTHOR]

Let = [ X 1 ⇉ X 0 ] be a Lie groupoid equipped with a connection, given by a smooth distribution ℋ ⊂ T X 1 transversal to the fibers of the source map. Under the assumption that the distribution ℋ is integrable, we define a version of de Rham cohomology for the pair (, ℋ) , and we study connections on principal G -bundles over (, ℋ) in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern–Weil theory and describe characteristic classes of principal G -bundles over a pair (, ℋ). [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero. We prove that the Brauer group of the moduli stack of stable parabolic PGL (r , k) -bundles on a smooth projective curve, with full flag quasi-parabolic structures at an arbitrary parabolic divisor, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic PGL (r , k) -bundles. We also compute the Brauer group of the smooth locus of this coarse moduli for more general quasi-parabolic types and weights satisfying certain conditions. [ABSTRACT FROM AUTHOR]

A constructive procedure is given to determine all ideals of a finite-dimensional solvable Lie algebra. This is used in determining all conjugacy classes of subalgebras of a given finite-dimensional solvable Lie algebra. [ABSTRACT FROM AUTHOR]

Let f : X → Y be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups f ∗ : π 1 et (X) → π 1 et (Y) is surjective. Fix a polarization on Y and equip X with the pullback, by f , of this polarization on Y. Given a stable vector bundle E on X , we prove that there is a vector bundle W on Y with f ∗ W isomorphic to E if and only if the direct image f ∗ E contains a stable vector bundle F such that degree (F) rank (F) = 1 degree (f) ⋅ degree (E) rank (E). We also prove that f ∗ V is stable for every stable vector bundle V on Y. [ABSTRACT FROM AUTHOR]

Given a compact connected Riemann surface Σ of genus g Σ ≥ 2 , and an effective divisor D = ∑ i n i x i on Σ with degree (D) < 2 (g Σ − 1) , there is a unique cone metric on Σ of constant negative curvature − 4 such that the cone angle at each point x i is 2 π n i [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on Σ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on Σ parametrized by a nonempty open subset of H 0 (Σ , K Σ ⊗ 2 ⊗ Σ (− 2 D)) that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case D = 0. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map ϕ: Symd(X)→Picd(X) is an embedding if d is less than the gonality of X. We investigate the curvature of the pull-back, by ϕ, of the flat metric on Picd(X). In particular, we show that when d = 1, the curvature is strictly negative everywhere if X is not hyperelliptic, and when X is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of X fixed by the hyperelliptic involution. [ABSTRACT FROM AUTHOR]

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space. [ABSTRACT FROM AUTHOR]

We study the following question: Given a vector bundle on a projective variety X such that the restriction of E to every closed curve C ⊂ X is ample, under what conditions E is ample? We first consider the case of an abelian variety X. If E is a line bundle on X , then we answer the question in the affirmative. When E is of higher rank, we show that the answer is affirmative under some conditions on E. We then study the case of X = G / P , where G is a reductive complex affine algebraic group, and P is a parabolic subgroup of G. In this case, we show that the answer to our question is affirmative if E is T -equivariant, where T ⊂ P is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on G / P. [ABSTRACT FROM AUTHOR]

We prove a Bochner-type vanishing theorem for compact complex manifolds Y in Fujiki class 𝒞 , with vanishing first Chern class, that admit a cohomology class [ α ] ∈ H 1 , 1 (Y , ℝ) which is numerically effective (nef) and has positive self-intersection (meaning ∫ Y α n > 0 , where n = dim ℂ Y). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold Y are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of Y must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold Y admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant. [ABSTRACT FROM AUTHOR]

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class 𝒞 , whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class 𝒞 and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type. [ABSTRACT FROM AUTHOR]

Let X be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer k ≥ 2 , we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle Ω X ⊗ k , where Ω X is the holomorphic cotangent bundle of X. Our first main result estimates the corresponding Bergman metric on X in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of X into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of Ω X ⊗ k . The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of X. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of Ω X ⊗ k and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on X. [ABSTRACT FROM AUTHOR]

Let E G be a Γ -equivariant algebraic principal G -bundle over a normal complex affine variety X equipped with an action of Γ , where G and Γ are complex linear algebraic groups. Suppose X is contractible as a topological Γ -space with a dense orbit, and x 0 ∈ X is a Γ -fixed point. We show that if Γ is reductive, then E G admits a Γ -equivariant isomorphism with the product principal G -bundle X × ρ E G (x 0) , where ρ : Γ → G is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal G -bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal G -bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math.27(14) (2016)]. [ABSTRACT FROM AUTHOR]

Given a compact Riemann surface and a moduli space of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on is parabolic stable with respect to a natural polarization on . [ABSTRACT FROM AUTHOR]

Let be a holomorphic principal -bundle on a compact connected Riemann surface , where is a connected reductive complex affine algebraic group. Fix a finite subset , and for each fix . Let be a maximal torus in the group of all holomorphic automorphisms of . We give a necessary and sufficient condition for the existence of a -invariant logarithmic connection on singular over such that the residue over each is . We also give a necessary and sufficient condition for the existence of a logarithmic connection on singular over such that the residue over each is , under the assumption that each is -rigid. [ABSTRACT FROM AUTHOR]

If is a semisimple Lie algebra of vector fields on with a split Cartan subalgebra , then it is proved here that the dimension of the generic orbit of coincides with the dimension of . As a consequence one obtains a local canonical form of in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates - for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on and to determine all representations of as local vector fields on . These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits as its symmetry algebra and for a system of two second-order ordinary differential equations that admits as its symmetry algebra. [ABSTRACT FROM AUTHOR]

Let be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let be a connected reductive affine algebraic group, defined over , such that is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal -bundles on determine uniquely the isomorphism class of . [ABSTRACT FROM AUTHOR]

Let be a complex abelian variety and a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal -bundles and -Higgs bundles on . We also describe the moduli spaces of -connections and the -character variety for . [ABSTRACT FROM AUTHOR]

We classify holomorphic as well as algebraic torus equivariant principal -bundles over a nonsingular toric variety , where is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results. [ABSTRACT FROM AUTHOR]

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov's sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group. [ABSTRACT FROM AUTHOR]

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor. [ABSTRACT FROM AUTHOR]

We construct a pair , where is a holomorphic vector bundle over a compact Riemann surface and a holomorphic subbundle, such that both and admit holomorphic connections, but does not. [ABSTRACT FROM AUTHOR]

Given a compact symplectic manifold M, with integral symplectic form, we prequantize a certain class of functions on the path space for M. The functions in question are induced by functions on M. We apply our construction to study the symplectic structure on the solution space of Klein-Gordon equation. [ABSTRACT FROM AUTHOR]

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by . Given a equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so. [ABSTRACT FROM AUTHOR]

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,...,L〉 → S with the determinant line bundle (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen-Mumford expansion, J. Differential Geom. 78 (2008) 475-496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles and 〈L,...,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,...,L〉 → S and is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91-96] between a Deligne paring and a certain determinant line bundle. [ABSTRACT FROM AUTHOR]

Given a group G and a class of manifolds 풞 (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold MG ∈ 풞 whose fundamental group is G. This article refines it: for a group G and a positive integer r find MG ∈ 풞 such that π1(MG) = G and πi(MG) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an MG ∈ 풞 can be found is called the homotopical height 풞(G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class 풞, leading to a hierarchy of these classes in terms of "softness" or "hardness" à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes 풮풫 and 풞풜 of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in 풮풫 and a manifold in 풞풜. For these classes, 풞(G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP. [ABSTRACT FROM AUTHOR]

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y --> X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When Xi is a Riemann surface of genus > 2, we show that any non-constant holomorphic map F : Y → X is of a special form. [ABSTRACT FROM AUTHOR]

We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle. [ABSTRACT FROM AUTHOR]

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic Ω/A-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle (pL*L*) ... (α*L), where α is the addition map on A x A, and pi is the projection of A x A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math. 22 (2011) 1529-1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction. [ABSTRACT FROM AUTHOR]

Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [Which 3-manifold groups are Kahler groups? J. Eur. Math. Soc. 11 (2009) 521-528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kahler groups. This gives a negative answer to a question of Gromov which asks whether Kähler groups can be characterized by their asymptotic geometry. [ABSTRACT FROM AUTHOR]

We give a criterion for the Kostant-Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra. [ABSTRACT FROM AUTHOR]

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where $M^2\cong {\mathcal{O}}_X$, together with the automorphisms induced by automorphisms of X. [ABSTRACT FROM AUTHOR]

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian-Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler. [ABSTRACT FROM AUTHOR]

Let A be a complex abelian variety. The moduli space ${\mathcal M}_C$ of rank one algebraic connections on A is a principal bundle over the dual abelian variety A∨ = Pic0(A) for the group $H^0(A, \Omega^1_A)$. Take any line bundle L on A∨; let ${\mathcal C}(L)$ be the algebraic principal $H^0(A^\vee, \Omega^1_{A^\vee})$-bundle over A∨ given by the sheaf of connections on L. The line bundle L produces a homomorphism $H^0(A, \Omega^1_A) \rightarrow H^0(A^\vee, \Omega^1_{A^\vee})$. We prove that ${\mathcal C}(L)$ is isomorphic to the principal $H^0(A^\vee, \Omega^1_{A^\vee})$-bundle obtained by extending the structure group of the principal $H^0(A, \Omega^1_A)$-bundle ${\mathcal M}_C$ using this homomorphism given by L. We compute the ring of algebraic functions on ${\mathcal C}(L)$. As an application of the above result, we show that ${\mathcal M}_C$ does not admit any nonconstant algebraic function, despite the fact that it is biholomorphic to (ℂ*)2 dim A implying that it has many nonconstant holomorphic functions. [ABSTRACT FROM AUTHOR]

We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E* is defined to be k-ample if the tautological line bundle $\mathcal{O}_{\mathbb{P}(E_{*})}(1)$ is k-ample. We establish some properties of parabolic k-ample bundles. [ABSTRACT FROM AUTHOR]

Let M be a C∞ manifold, and let ${\mathcal P}M$ be the space of all smooth maps from [0, 1] to M. We investigate geometric structures on ${\mathcal P}M$ constructed from the geometric structures on M. In particular, we show that a generalized (almost) complex structure on M produce a generalized (almost) complex structure on ${\mathcal P}M$. [ABSTRACT FROM AUTHOR]

Let EG be a stable principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein-Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL(n, ℂ). If $K_M^{-1}$ is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler-Einstein metric, then the algebraic holonomy of T1, 0M is GL(n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183-211. [ABSTRACT FROM AUTHOR]

Let E* be a parabolic vector bundle over a smooth complex projective curve. We prove that E* admits an algebraic connection if and only if the parabolic degree of every parabolic vector bundle which is a direct summand of E* is zero. In particular, all parabolic semistable vector bundles of parabolic degree zero admit an algebraic connection. [ABSTRACT FROM AUTHOR]

There are two families of geometric structures associated to a surface, with both structures related to representations of the fundamental group of the surface into SL(2, ℂ). These are projective structures on the surface, and Higgs bundles for a given conformal structure of the surface. This note discusses the links between the two. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and G/Z(G) = \prod_{i=1}^m G_i$, where each Gi is simple with trivial center. For i ∈ [1, m], let ρi : G → Gi be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρi(G) ⊂ Gi is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let EG → M be a principal G-bundle. We prove that the following six statements are equivalent: (1) The line bundle EG(χ) → EG/P associated to the principal P-bundle EG → EG/P for the character χ is numerically effective. (2) The sequence of principal G-bundles $\{(F^a_M)^*E_G\}_{a\geq 0}$ is bounded, where FM is the absolute Frobenius morphism of M. (3) The principal G-bundle EG is strongly semistable with respect to ξ, and c2(ad(EG)) is numerically equivalent to zero. (4) The principal G-bundle EG is strongly semistable with respect to ξ, and [c2(ad(EG))c1(ξ)d-2] = 0. (5) The adjoint vector bundle ad(EG) is numerically effective. (6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*EG → Y is semistable. [ABSTRACT FROM AUTHOR]

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad(EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let $\mathcal{M}_\xi(r)$ denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on $\mathcal{M}_\xi(r)$ by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let $\mathcal{M}_H$ denote the moduli space of semistable Higgs Gp(2n, ℂ)-bundles over X of fixed topological type. The complex variety $\mathcal{M}_H$ has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilbℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilbℓ(KX) with the symplectic form on $\mathcal{M}_H$. [ABSTRACT FROM AUTHOR]

The stratified vector bundles on a smooth variety defined over an algebraically closed field k form a neutral Tannakian category over k. We investigate the affine group-scheme corresponding to this neutral Tannakian category. [ABSTRACT FROM AUTHOR]

We establish a Kobayashi–Hitchin correspondence for the stable Higgs sheaves on a compact Kähler manifold. Using it, we also obtain a Kobayashi–Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group. [ABSTRACT FROM AUTHOR]

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2(ad(EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover $\widehat{\varpi} : \widetilde{M} \rightarrow M$ and a reduction of structure group $\widehat{E}_B \subset \widehat{\varpi}^*E_G$ to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent. [ABSTRACT FROM AUTHOR]